Rotor-Router Aggregation on the Comb

Understand

We prove a shape theorem for rotor-router aggregation on the comb, for a specific initial rotor configuration and clockwise rotor sequence for all vertices.

• Furthermore, as an application of rotor-router walks, we describe the harmonic measure of the rotor-router aggregate and related shapes, which is useful in the study of other growth models on the comb.
• We also identify the shape for which the harmonic measure is uniform.
• This gives the first known example where the rotor-router cluster has non-uniform harmonic measure, and grows with different speeds in different directions.